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Theorem elcncf1di 13928
Description: Membership in the set of continuous complex functions from 
A to  B. (Contributed by Paul Chapman, 26-Nov-2007.)
Hypotheses
Ref Expression
elcncf1d.1  |-  ( ph  ->  F : A --> B )
elcncf1d.2  |-  ( ph  ->  ( ( x  e.  A  /\  y  e.  RR+ )  ->  Z  e.  RR+ ) )
elcncf1d.3  |-  ( ph  ->  ( ( ( x  e.  A  /\  w  e.  A )  /\  y  e.  RR+ )  ->  (
( abs `  (
x  -  w ) )  <  Z  -> 
( abs `  (
( F `  x
)  -  ( F `
 w ) ) )  <  y ) ) )
Assertion
Ref Expression
elcncf1di  |-  ( ph  ->  ( ( A  C_  CC  /\  B  C_  CC )  ->  F  e.  ( A -cn-> B ) ) )
Distinct variable groups:    x, w, y, A    w, B, x, y    w, F, x, y    ph, w, x, y   
w, Z
Allowed substitution hints:    Z( x, y)

Proof of Theorem elcncf1di
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 elcncf1d.1 . . 3  |-  ( ph  ->  F : A --> B )
2 elcncf1d.2 . . . . . 6  |-  ( ph  ->  ( ( x  e.  A  /\  y  e.  RR+ )  ->  Z  e.  RR+ ) )
32imp 124 . . . . 5  |-  ( (
ph  /\  ( x  e.  A  /\  y  e.  RR+ ) )  ->  Z  e.  RR+ )
4 an32 562 . . . . . . . . 9  |-  ( ( ( x  e.  A  /\  w  e.  A
)  /\  y  e.  RR+ )  <->  ( ( x  e.  A  /\  y  e.  RR+ )  /\  w  e.  A ) )
54anbi2i 457 . . . . . . . 8  |-  ( (
ph  /\  ( (
x  e.  A  /\  w  e.  A )  /\  y  e.  RR+ )
)  <->  ( ph  /\  ( ( x  e.  A  /\  y  e.  RR+ )  /\  w  e.  A ) ) )
6 anass 401 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  e.  A  /\  y  e.  RR+ ) )  /\  w  e.  A
)  <->  ( ph  /\  ( ( x  e.  A  /\  y  e.  RR+ )  /\  w  e.  A ) ) )
75, 6bitr4i 187 . . . . . . 7  |-  ( (
ph  /\  ( (
x  e.  A  /\  w  e.  A )  /\  y  e.  RR+ )
)  <->  ( ( ph  /\  ( x  e.  A  /\  y  e.  RR+ )
)  /\  w  e.  A ) )
8 elcncf1d.3 . . . . . . . 8  |-  ( ph  ->  ( ( ( x  e.  A  /\  w  e.  A )  /\  y  e.  RR+ )  ->  (
( abs `  (
x  -  w ) )  <  Z  -> 
( abs `  (
( F `  x
)  -  ( F `
 w ) ) )  <  y ) ) )
98imp 124 . . . . . . 7  |-  ( (
ph  /\  ( (
x  e.  A  /\  w  e.  A )  /\  y  e.  RR+ )
)  ->  ( ( abs `  ( x  -  w ) )  < 
Z  ->  ( abs `  ( ( F `  x )  -  ( F `  w )
) )  <  y
) )
107, 9sylbir 135 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  A  /\  y  e.  RR+ ) )  /\  w  e.  A
)  ->  ( ( abs `  ( x  -  w ) )  < 
Z  ->  ( abs `  ( ( F `  x )  -  ( F `  w )
) )  <  y
) )
1110ralrimiva 2550 . . . . 5  |-  ( (
ph  /\  ( x  e.  A  /\  y  e.  RR+ ) )  ->  A. w  e.  A  ( ( abs `  (
x  -  w ) )  <  Z  -> 
( abs `  (
( F `  x
)  -  ( F `
 w ) ) )  <  y ) )
12 breq2 4006 . . . . . 6  |-  ( z  =  Z  ->  (
( abs `  (
x  -  w ) )  <  z  <->  ( abs `  ( x  -  w
) )  <  Z
) )
1312rspceaimv 2849 . . . . 5  |-  ( ( Z  e.  RR+  /\  A. w  e.  A  (
( abs `  (
x  -  w ) )  <  Z  -> 
( abs `  (
( F `  x
)  -  ( F `
 w ) ) )  <  y ) )  ->  E. z  e.  RR+  A. w  e.  A  ( ( abs `  ( x  -  w
) )  <  z  ->  ( abs `  (
( F `  x
)  -  ( F `
 w ) ) )  <  y ) )
143, 11, 13syl2anc 411 . . . 4  |-  ( (
ph  /\  ( x  e.  A  /\  y  e.  RR+ ) )  ->  E. z  e.  RR+  A. w  e.  A  ( ( abs `  ( x  -  w ) )  < 
z  ->  ( abs `  ( ( F `  x )  -  ( F `  w )
) )  <  y
) )
1514ralrimivva 2559 . . 3  |-  ( ph  ->  A. x  e.  A  A. y  e.  RR+  E. z  e.  RR+  A. w  e.  A  ( ( abs `  ( x  -  w
) )  <  z  ->  ( abs `  (
( F `  x
)  -  ( F `
 w ) ) )  <  y ) )
161, 15jca 306 . 2  |-  ( ph  ->  ( F : A --> B  /\  A. x  e.  A  A. y  e.  RR+  E. z  e.  RR+  A. w  e.  A  ( ( abs `  (
x  -  w ) )  <  z  -> 
( abs `  (
( F `  x
)  -  ( F `
 w ) ) )  <  y ) ) )
17 elcncf 13922 . 2  |-  ( ( A  C_  CC  /\  B  C_  CC )  ->  ( F  e.  ( A -cn-> B )  <->  ( F : A --> B  /\  A. x  e.  A  A. y  e.  RR+  E. z  e.  RR+  A. w  e.  A  ( ( abs `  ( x  -  w
) )  <  z  ->  ( abs `  (
( F `  x
)  -  ( F `
 w ) ) )  <  y ) ) ) )
1816, 17syl5ibrcom 157 1  |-  ( ph  ->  ( ( A  C_  CC  /\  B  C_  CC )  ->  F  e.  ( A -cn-> B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    e. wcel 2148   A.wral 2455   E.wrex 2456    C_ wss 3129   class class class wbr 4002   -->wf 5210   ` cfv 5214  (class class class)co 5871   CCcc 7805    < clt 7987    - cmin 8123   RR+crp 9648   abscabs 10998   -cn->ccncf 13919
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4120  ax-pow 4173  ax-pr 4208  ax-un 4432  ax-setind 4535  ax-cnex 7898
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4003  df-opab 4064  df-id 4292  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-iota 5176  df-fun 5216  df-fn 5217  df-f 5218  df-fv 5222  df-ov 5874  df-oprab 5875  df-mpo 5876  df-map 6646  df-cncf 13920
This theorem is referenced by:  elcncf1ii  13929  cncfmptc  13944  cncfmptid  13945  addccncf  13948  negcncf  13950
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